When we encounter an exercise in which there is a root applied to another root, we will multiply the order of the first root by the order of the second and the order obtained (the product of both) will be raised as a root in our number (as generally power of a power) Let's put it this way:Β Β
In the blog ofTutorelayou will find a variety of articles about mathematics.
Examples and exercises with solutions on root extraction
Exercise #1
Solve the following exercise:
2ββ=
Video Solution
Step-by-Step Solution
To solve 2ββ, we will use the property of roots.
Step 1: Recognize that 2ββ involves two square roots.
Step 2: Each square root can be expressed using exponents: 2β=21/2.
Step 3: Therefore, 2ββ=(21/2)1/2.
Step 4: Apply the formula for the root of a root: (xa)b=xab.
Step 5: For (21/2)1/2, this means we compute the product of the exponents: (1/2)Γ(1/2)=1/4.
Step 6: The expression simplifies to 21/4, which is written as 42β.
Therefore, 2ββ=42β.
This corresponds to choice 2: 42β.
The solution to the problem is 42β.
Answer
42β
Exercise #2
Solve the following exercise:
535ββ=
Video Solution
Step-by-Step Solution
To solve the problem of finding 535ββ, we'll use the formula for a root of a root, which combines the exponents:
Step 1: Express each root as an exponent.
We start with the innermost root: 35β=51/3.
Step 2: Apply the outer root.
The square root to the fifth power is expressed as: 551/3β=(51/3)1/5.
Step 3: Combine the exponents.
Using the exponent rule (am)n=amΓn, we get: (51/3)1/5=5(1/3)Γ(1/5)=51/15.
Step 4: Convert the exponent back to root form.
This can be written as 155β.
Therefore, the simplified expression of 535ββ is 155β.
Answer
155β
Exercise #3
Solve the following exercise:
10101ββ=
Video Solution
Step-by-Step Solution
To solve this problem, we'll observe the following process:
Step 1: Recognize the expression 10101ββ involves nested roots.
Step 2: Apply the formula for nested roots: nmxββ=nβ mxβ.
Step 3: Set n=10 and m=10, resulting in 10Γ101β=1001β.
Step 4: Simplify 1001β. Any root of 1 is 1, as 1k=1 for any positive rational number k.
Thus, the evaluation of the original expression 10101ββ equals 1.
Comparing this result to the provided choices:
Choice 1 is 1.
Choice 2 is 1001β, which is also 1.
Choice 3 is 1β=1.
Choice 4 states all answers are correct.
Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.
Thus, the correct selection is: All answers are correct.
Answer
All answers are correct.
Exercise #4
Solve the following exercise:
62ββ=
Video Solution
Step-by-Step Solution
Express the definition of root as a power:
naβ=an1β
Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown below:
n=2
Meaning:
aβ=2aβ=a21β
Now convert the roots in the problem using the root definition provided above. :
62ββ=6221ββ=(221β)61β
In the first stage we applied the root definition as a power mentioned earlier to the inner expression (meaning inside the larger-outer root) and then we used parentheses and applied the same definition to the outer root.
Let's recall the power law for power of a power:
(am)n=amβ n
Apply this law to the expression that we obtained in the last stage:
In the first stage we applied the power law mentioned above and then proceeded first to simplify the resulting expression and then to perform the multiplication of fractions in the power exponent.
Let's summarize the various steps of the solution thus far:
62ββ=(221β)61β=2121β
In the next stage we'll apply once again the root definition as a power, (that was mentioned at the beginning of the solution) however this time in the opposite direction:
an1β=naβ
Let's apply this law in order to present the expression we obtained in the last stage in root form:
2121β=122β
We obtain the following result: :
62ββ=2121β=122β
Therefore the correct answer is answer A.
Answer
122β
Exercise #5
Solve the following exercise:
8ββ=
Video Solution
Step-by-Step Solution
In order to solve the given problem, we'll follow these steps:
Step 1: Convert the inner square root to an exponent: 8β=81/2.
Step 2: Apply the root of a root property: 8ββ=(8β)1/2=(81/2)1/2.
Step 3: Simplify the expression using exponent rules: (81/2)1/2=8(1/2)β (1/2)=81/4.
The nested root expression simplifies to 81/4.
Therefore, the simplified expression of 8ββ is 841β.
After comparing this result with the multiple choice answers, choice 2 is correct.
Answer
841β
Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today